| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·11-s + 13-s − 2·19-s − 2·21-s + 2·23-s + 4·27-s + 2·29-s − 8·33-s + 2·37-s − 2·39-s + 4·43-s + 49-s + 4·53-s + 4·57-s − 6·59-s − 6·61-s + 63-s − 4·67-s − 4·69-s + 6·71-s + 14·73-s + 4·77-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.458·19-s − 0.436·21-s + 0.417·23-s + 0.769·27-s + 0.371·29-s − 1.39·33-s + 0.328·37-s − 0.320·39-s + 0.609·43-s + 1/7·49-s + 0.549·53-s + 0.529·57-s − 0.781·59-s − 0.768·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s + 0.712·71-s + 1.63·73-s + 0.455·77-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045305266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045305266\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.32974134969706, −12.72261948819446, −12.22151913626641, −12.01122893059461, −11.32625620404725, −11.16543126118582, −10.60188591051923, −10.21520656432154, −9.461839429505823, −9.034813563082897, −8.642357669990535, −7.936556918648524, −7.487123960021888, −6.711707633311942, −6.443200880577397, −6.037475734099282, −5.417013319154256, −4.882559213881308, −4.444503714019263, −3.837847615067045, −3.255926397974691, −2.435896073449769, −1.742772180702764, −1.015345914370140, −0.5534799627508702,
0.5534799627508702, 1.015345914370140, 1.742772180702764, 2.435896073449769, 3.255926397974691, 3.837847615067045, 4.444503714019263, 4.882559213881308, 5.417013319154256, 6.037475734099282, 6.443200880577397, 6.711707633311942, 7.487123960021888, 7.936556918648524, 8.642357669990535, 9.034813563082897, 9.461839429505823, 10.21520656432154, 10.60188591051923, 11.16543126118582, 11.32625620404725, 12.01122893059461, 12.22151913626641, 12.72261948819446, 13.32974134969706
